Water temperature has no relation to flushing away hair.

Washers nowadays may not handle pet hair very well, particularly low-water HE machines. Expect that the dryer will do more to clear it away.

Try drying with a softener sheet which may help reduce static so the hair can be pulled off the fabric into the dryer's filter.

Dadoes is right. Newer washing machines (FL) don't have lint filters and depend on the dryer for lint/pet hair removal.
We have four large dogs. When we wash their dog beds there is at least 1/2 inch coating of hair floating on top of the water while washing(TL machine). I use a 4inch screen filter that is concave and just skim the top of the water with it. All the hair eventually gets picked up and we never see any in the dryer.
Or you could find a vintage GE Filter Flow machine, those filtered the wash water while washing and would get every last bit of hair out of the wash!

Haven't tried it yet myself, but on another board it was recommended to tumble dry first for 15 minutes or so, then wash as normal, and then a final tumble dry. (Making sure to clean out the lint filter between sessions.

the dryer is your only hope for hair removal. try running it on "tumble only" for quite a while after everything is dry. the washer won't remove hair.